Recalibration-Aware Optimization of ZEFOZ Operating Points in $^{167}\mathrm{Er}^{3+}:\mathrm{Y}_2\mathrm{SiO}_5$ for Quantum Memories

Abstract

Erbium-doped solids like $^{167}\mathrm{Er}^{3+}:\mathrm{Y}_2\mathrm{SiO}_5$ have properties that make them suitable for quantum-network memories, however, their unquenched effective electronic spin causes rapid magnetic decoherence. Operating at Zero First-Order Zeeman (ZEFOZ) points allows us to mitigate this, where we evaluate the ZEFOZ landscape for Site-2 $^{167}\mathrm{Er}^{3+}$ by incorporating a full-Hessian spin-Hamiltonian treatment with the anisotropic magnetic-field covariance derived from the native $^{89}\mathrm{Y}$ crystal lattice directly. Operating-point screening criteria were adopted to devise a Pareto-optimal frontier of four transition pairs to balance modeled Y-only magnetic pure-dephasing lifetimes ($T_{ϕ,Y}^{(2)}$) against microwave control authority. The $ε_{14}\leftrightarrowε_{15}$ transition exhibits the longest modeled Y-bath-induced quadratic pure-dephasing time, $T_{ϕ,Y}^{(2)}>2.2\times10^5~\mathrm{s}$, while the $ε_{12}\leftrightarrowε_{13}$ transition stood out, achieving a Rabi frequency coefficient of $1.248 \times 10^5~\mathrm{Hz/mT}$. An assumption-based Hamiltonian-uncertainty analysis demonstrates that while uncompensated deviations shift the ZEFOZ condition away from its nominal magnetic-field coordinate, the shifted operating points can be recovered via numerical vector-field recalibration. The Pareto-optimal coordinates were mathematically recovered for all sampled Hamiltonian perturbations, and the $84.1^{\mathrm{st}}$-percentile field relocations ranged from $10.95$ to $27.97~\mathrm{mT}$. The intended operation near $10~\mathrm{mK}$ must be taken into account for the suppression of Er-induced spectral diffusion through electronic-spin polarization, while the extended Y-bath pure-dephasing timescales also remain conditional on the adopted temporal-correlation model.